An Extension of Casson's Invariant. (AM-126), Volume 126

By: Kevin Walker

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Paperback | 2 June 1992

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150 Pages

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23.4 x 15.2 x 1.27

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This book describes an invariant, l, of oriented rational homology 3-spheres which is a generalization of work of Andrew Casson in the integer homology sphere case. Let R(X) denote the space of conjugacy classes of representations of p(X) into SU(2). Let (W,W,F) be a Heegaard splitting of a rational homology sphere M. Then l(M) is declared to be an appropriately defined intersection number of R(W) and R(W) inside R(F). The definition of this intersection number is a delicate task, as the spaces involved have singularities.

A formula describing how l transforms under Dehn surgery is proved. The formula involves Alexander polynomials and Dedekind sums, and can be used to give a rather elementary proof of the existence of l. It is also shown that when M is a Z-homology sphere, l(M) determines the Rochlin invariant of M.

Industry Reviews

"[This is] a monograph describing Walker's extension of Casson's invariant to Q HS ... This is a fascinating subject and Walker's book is informative and well written ... it makes a rather pleasant introduction to a very active area in geometric topology."--Bulletin of the American Mathematical Society

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